Testing the Anti-cancer Drug, Glofitamab, in Patients With Mantle Cell Lymphoma (A Type of Blood Cancer) Whose Disease Returned After CAR-T Cell Therapy

Inclusion Criteria: * Patients must have histologically or cytologically confirmed diagnosis of mantle cell lymphoma that is relapsed (disease progression after the last treatment regimen) or refractory (failure to achieve a partial or complete response from the last treatment regimen) * Patients must have been previously treated with an anti-CD19 CAR T-cell therapy and have failed or been intolerant to Bruton's tyrosine kinase (BTK) inhibition. Both commercial and investigational CAR-T products which target CD19 will be allowed, including dual-targeting products * Patients must have at least one bi-dimensionally measurable (defined as at least 1.5 cm) nodal lesion or one bi-dimensionally measurable (at least 1 cm) extranodal lesion, as measured on CT scan * Age ≥ 18 years. Because no dosing or adverse event (AE) data are currently available on the use of glofitamab and obinutuzumab in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 (Karnofsky ≥ 60%) * Absolute neutrophil count ≥ 1,000/mcL * Platelets ≥ 50,000/mcL * Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) or ≤ 3 x institutional ULN if the patient has Gilbert syndrome * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3 x institutional ULN * Creatinine ≤ 1.5 x institutional ULN OR glomerular filtration rate (GFR) ≥ 60 mL/min/1.73 m\^2 * Patients with human immunodeficiency virus (HIV) infection are eligible if on effective anti-retroviral therapy with undetectable viral load within 6 months * Patients with a history of hepatitis B virus (HBV) infection or positive total hepatitis B core antibody (HBcAb) are eligible if the hepatitis B surface antigen (HBsAg) is negative and HBV DNA viral load is undetectable by polymerase chain reaction (PCR) at the time of screening. Such patients must be managed with appropriate anti-viral therapy, if indicated, and must be willing to undergo HBV DNA testing on day 1 of each cycle and every 3 months for at least 12 months after the final cycle of study treatment * Patients with a history of hepatitis C virus (HCV) infection or positive HCV antibody are eligible if HCV ribonucleic acid (RNA) viral load is undetectable by PCR * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression and patients are asymptomatic from CNS involvement * Patients with new, progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that patients are asymptomatic and immediate CNS specific treatment is not required and is unlikely to be required during the first cycle of therapy * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial, such as patients with prostate cancer or breast cancer receiving hormonal therapy * The effects of glofitamab and obinutuzumab on the developing human fetus are unknown. For this reason and because glofitamab and obinutuzumab are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 2 months after completion of glofitamab and 6 months after completion of obinutuzumab administration. Women of childbearing potential must use effective contraceptive precautions 2 months after completion of glofitamab treatment and 18 months after the last dose of obinutuzumab treatment * Ability to understand and the willingness to sign a written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of study participants Exclusion Criteria: * Patients who have not recovered from AEs due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia * Patients who are receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to glofitamab and obinutuzumab * Pregnant women are excluded from this study because glofitamab and obinutuzumab are agents with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for AEs in nursing infants secondary to treatment of the mother with glofitamab and obinutuzumab, breastfeeding should be discontinued if the mother is treated with glofitamab and obinutuzumab * Patients with active CRS or requiring intervention for CRS within 14 days prior to study enrollment * Patients with active neurotoxicity or requiring intervention for neurotoxicity within 14 days prior to study enrollment * Patients requiring antimicrobial treatment or hospitalization for active infection within 14 days prior to study enrollment for known active bacterial, viral (including severe acute respiratory syndrome coronavirus 2 \[SARS-CoV-2\]), fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds). Infection must be completely resolved prior to initiation of study treatment * Patients receiving systemic immunosuppressive medications within 14 days prior to study enrollment, including, but not limited to: prednisone ≥ 20 mg per day, azathioprine, methotrexate, anti-tumor necrosis factor agents, anti-i